The aim of this study is to provide a class of state observers, based on differential neural networks, to approximate a class of advanced oxidation systems, based on the application of ozone high oxidant power and catalyst (the named catalytic ozonation). The study considers the design of a state observer for uncertain systems with the restrictions of the ozonation system, including the positivity of the states, as well as the control action. The observer includes a projection operator which is motivated by the state constraints. The learning laws of the proposed differential neural networks are obtained using a class of controlled state restricted Lyapunov functions. The detailed stability analysis proves the input to state stability with respect to the modeling error, as well as the bounded uncertainties of the ozonation system. The experimental confirmation of the state estimation is also presented. The experimental case considers the ozonation of a toxic organic contaminant (therephtalic acid) which is a regular pollutant of the plastic industry wastewater.